MAGIC014: Hydrodynamic Stability Theory

This is offered as a core course for Applied.

It will be assumed that students are familiar with the Navier-Stokes equations.

Any previous experience of perturbation methods would be an advantage, but is not essential, as the main ideas will be introduced as needed. 

0. Some pictures of unstable flows (motivation)
1. Introduction The idea of instability(Approximately) parallel shear flows - e.g. pipe flow, boundary layers, channel flows, jets, wakes, mixing layersShear layer stability equations - reduction to linear ODEs
2. Inviscid stability theory Stability theorems - inflexion points, etc.Piecewise-linear profilesCritical points - Tollmien's solutionsEmergence of layered structures in the long-wave limitMatched asymptotic expansionsSecond order long-wave theory capturing critical layers
3. Viscous stability theory Thin viscous layers within inviscid flowDestabilizing effects of viscosityAn interpretation of the viscous instability mechanism
4. Weakly nonlinear theory Solvability conditions - when do solutions to forced equations exist?Higher order expansions in the amplitude parameter.Multiple-scales theory.Amplitude equations - supercritical/subcritical bifurcations.Wave interactions - resonant and nonresonant cases.
5. Absolute and convective instabilities Upstream and downstream propagation.Initial value problems.Saddle point methods.